Let \(\mathcal {I}(\Xi )\subset \Pi ^2\)  be the vanishing ideal of \(\Xi \) 
and \(\mathcal {G}_\prec (\Xi )\)  the reduced Gröbner basis for
\(\mathcal {I}(\Xi )\)  w.r.t.\(\prec \) . In 1982, [1]} proposed
Buchberger-Möller(BM for short) algorithm for computing bases of
vanishing ideals. Specifically, with inputted \(\Xi \)  and \(\prec \) , BM
algorithm outputs \(\mathcal {G}_\prec (\Xi )\) ,
\(\mathcal {N}_\prec (\Xi )\) , and a DRINB for Lagrange interpolation
(REF ), namely arbitrary bivariate Lagrange interpolation
can be solved with it algorithmly. Nonetheless, the poor complexity
of BM algorithm(see [2]}) severely constrains its
applications.
